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7 SPACES OF CONTINUOUS MULTILINEAR MAPPINGS 107
For each u e J*?(E; F), let \\u\\ be the g.l.b. of all constants a > 0 which
satisfy the relation \\u(x)\\ < a • ||jc|| (see (5.5.1)) for all x. We can also write
(5.7.1) |H= sup ||w(x)||.
ll*ll<i
For by definition, for each a > \\u\\9 and \\x\\ < 1, \\u(x)\\ *^a, hence
sup \\u(x)\\ < \\u\\; this already proves (5.7.1) for \\u\\ = 0. If ||t/||>0,
HJC||^I
for any b such that 0 < b < \\u\\, there is an x e E such that \\u(x)\\ > b\\x\\;
this implies x^O, hence if z = x/||x||, we still have \\u(z)\\ >b • ||z|| = b, and as ||z|| = 1, this proves that b < sup ||H(X)||, hence \\u\\ < sup ||w(x)||,
ll*ll*i 11*11*1
and (5.7.1) is proved. The same argument also shows that if E ^ {0}
(5.7.2) Hull = sup ||«(z)||.
11*11 = 1
We now show that \\u\\ is a norm on the vector space jSf (E; F). For if
u = 0, then \\u\\ = 0 by (5.7.1), and conversely if \\u\\ = 0, then u(x) = 0 for ||jc|| < 1, hence, for any x ^ 0 in E, u(x) = ||x||i/(jc/||x||) = 0. It also follows from (5.7.1) that \\Au\\ = |A|-||i/||; finally, if w = u + v, we have \\w(x)\\ < ||u(x)|| + \\v(x)l hence ||w|| ^ ||W|| + |M| from (5.7.1). |
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(5.7.3) If¥ is complete, so is the normed space &(E; F).
For let (un) be a Cauchy sequence in J$?(E; F); for any e > 0, there is
therefore an n0 such that ||WOT — un\\ <e for m^n0y n^nQ. By (5.7.1), for any x such that ||x|| < 1, we therefore have ||wm(^) — un(x)\\ < e for m ^ n0, n^t n0; this shows that the sequence (wn(x)) is a Cauchy sequence in F, hence converges to an element v(x) 6 F. This is also true for any x e E, since we can write x = Iz with ||z|| < 1, hence un(x) = lun(z) tends to a limit z;(,x) = h)(z). From the relation un(x + j) = wn(^) + ww(y) and from (5.1.5) it follows that u(;c + y) = v(x) + ^(>?), and one shows similarly that y(Ajc) = A,v(x), in other words v in linear. Finally, from ||ww(.x) — un(x)\\ < e for m^«0, n^n0t we deduce ||i;(;c) — wn(x)|| < e for ||jc|| <1, hence ||y(jc)|| < ||wn|| + e, which proves (by (5.5.1)) that v is continuous, hence in &(E; F); furthermore ||z? - wj| *S e for n ^ n0 (by (5.7.1)), which proves the sequence (un) converges to v.
From the definition it follows that, for every x e E and every u e «£? (E; F),
(5.7.4) IM*)II<NNMI
which proves that the bilinear mapping (x, w) -> w(x) of E x «Sf (E; F) into
F is continuous (by (5.5.1)). |
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